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Use the principle of mathematical induction to prove, for all n∈N, 102n–1+1 is divisible by 11.

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Let the given statement 

T(n) = 102n–1+1 be a multiple of 11 

⇒ 102n–1+1 = M(11)

Basic Step: 

For n = 1, 102×1–1+1 = 10+1 = 11 which is divisible by 11. 

Induction Step: 

Assume that T(k) = 102k–1+1 is divisible by 11. 

⇒ 102k–1+1 = M(11)  ∀  n∈N .....(i) 

Then, we now show that T(k + 1) is true. 

T(k + 1) = 102(k +1)–1+1 = 102k–1+2 + 1 = 102 . 102k–1+1 

= 100 (M (11) – 1) + 1 (From (i)) 

= 100 . M(11) – 100 + 1 = 100 . M (11) – 99 

⇒ T(k + 1) is divisible by 11, when T(k) is divisible by 11. 

⇒ T(n) holds true for all n∈N.

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